| L(s) = 1 | + 32·17-s + 428·25-s − 1.24e3·41-s − 476·49-s − 984·73-s − 5.56e3·89-s − 1.20e3·97-s − 6.20e3·113-s + 1.74e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.98e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 0.456·17-s + 3.42·25-s − 4.75·41-s − 1.38·49-s − 1.57·73-s − 6.63·89-s − 1.26·97-s − 5.16·113-s + 1.30·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.63·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.260274951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260274951\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 214 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 34 p T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 870 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3994 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6550 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4338 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46662 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 59134 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 74410 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 312 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 20186 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 178974 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 226998 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 346246 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 436538 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 343478 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 257070 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 246 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 931870 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 195606 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 1392 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 302 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.70452409333778366591594433586, −6.53481314099422948401845256461, −6.49269611964710388689063653938, −5.65347616855220775776091121568, −5.59942407148581592771416956372, −5.56590742878226441953638798846, −5.40490202981920661751287285999, −4.88867339007516906535949521292, −4.85134172251848842998253374235, −4.55380884912313213330433087619, −4.54520414980913956935308001048, −3.91959040054945740303870633145, −3.77781344438274703600685602465, −3.67275405902544004804967415232, −3.11875767581333603192759740542, −2.96907258457680943611021892785, −2.82864035433359919397687291548, −2.70417249903300656061131754117, −2.24457335923841451895262156729, −1.51086005701955343445194400258, −1.50701856831571009167156852478, −1.45568381472207773888265977932, −1.10105881699621432751345220702, −0.39322742959378937911642672746, −0.17824465339069045753305772108,
0.17824465339069045753305772108, 0.39322742959378937911642672746, 1.10105881699621432751345220702, 1.45568381472207773888265977932, 1.50701856831571009167156852478, 1.51086005701955343445194400258, 2.24457335923841451895262156729, 2.70417249903300656061131754117, 2.82864035433359919397687291548, 2.96907258457680943611021892785, 3.11875767581333603192759740542, 3.67275405902544004804967415232, 3.77781344438274703600685602465, 3.91959040054945740303870633145, 4.54520414980913956935308001048, 4.55380884912313213330433087619, 4.85134172251848842998253374235, 4.88867339007516906535949521292, 5.40490202981920661751287285999, 5.56590742878226441953638798846, 5.59942407148581592771416956372, 5.65347616855220775776091121568, 6.49269611964710388689063653938, 6.53481314099422948401845256461, 6.70452409333778366591594433586
